Introduction to Quantum Mechanics (4).pdf

2
Start of thinking of science
Supposed to resolve almost
all the problems
Applied to
all the systems.
Transition from old to
modern era:
Elementary quantum mechanics
Particles and waves
Quantum mechanics
Newtonian mechanics
Applied to
visible systems
Start of formulating science
Supposed to resolve all problems
in chemistry, almost all problems
of physics
Philosophical
approach
Probabilistic
approach
State of the particle: position and momentum
Dual nature
(Quantum) State of the system! wave function

3
Without Quantum Mechanics, we could
never have designed and built:
semiconductor devices
computers, cell phones, etc.
Lasers, CD/DVD players,
MRI technology,
Nuclear reactors,
Atomic clocks (GPS navigation)

4
Quantum theory
Optics
Nuclear
physics Atoms and
molecules
Subatomic
particles
Medical
uses
Power and
bombs
Materials and
technology
Evolution
of universe
LASER Communication
Quantum
cryptography
Quantum
computer

5
Size
Speed
Classical
mechanics
Relativistic
mechanics
Relativistic
Q M
Quantum
mechanics
c
Relativistic
Domain
Microscopic
Domain

6
Crises in
physics that
demanded
Quantum
Mechanics
Blackbody
radiation
Photoelectric
effect
Atomic
structure

8
Photoelectric effect
Metals would shed electrons when
certain light was incident on it.
1887, Heinrich Hertz
Wave model of light
- increasing light intensity would increase the
kinetic energy of emitted photoelectrons,
(Brightness should produce more electrons!)
- increasing the frequency would increase
measured current.

9
Experiments showed that
- increasing the light frequency
increased the kinetic energy of
the photoelectrons,
- increasing the light amplitude
increased the current.
It has already been proved that light is a wave.
Solids are made by binding of atoms and atoms
contain electrons.
So, why only certain light is able to generate
electrons, and why not all sorts of light?

10
Photoelectric effect: three challenges
• There is no time interval b/w the arrival of light at a metal surface and the
emission of photoelectrons. How ever, because the energy in an em
wave is supposed to be spread across the wavefronts, a period of time
should elapse before an individual electron accumulates enough energy
to leave the metal.
• A bright light yields more photoelectrons than a dim one of the same
frequency, but the electron energies remain the same. The em theory of
light, on the con trary, predicts that the more intense the light, the greater
the energies of the electrons.
• The higher the frequency of the light, the more energy the photoelectrons
have. At frequencies below a certain critical frequency 0, which is
characteristic of each particular metal, no electrons are emitted.

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)
Heisenberg: Uncertainty
in measurements (1923)

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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Planck’s consideration:
the energy exchange
between radiation and
matter must be discrete.
The radiation (of frequency ν) emitted by the
matter (from the walls of the blackbody)
must come only in integer multiples of hν.
Eν = nhν, n = 1, 2, 3, 4 …
Rayleigh’s Classical
assumption:
radiation can exchange
any amount (continuum) of
energy with matter.
<E> = kBT

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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If hν is larger than the metal’s work
function W, the electron will then be
knocked out of the metal. Hence no
electron can be emitted from the metal’s
surface unless hν > W.
• Einstein thought that light is
equivalent to wave packets
called photons – particle nature.
• Energy of each packet is hν. ‘h’
is the Planck’s constant.
When a photon of frequency ν is incident
on a metal, it is entirely absorbed by the
electron near the surface.

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(a)The wave theory of light explains
diffraction and interference.
(b)The quantum theory explains the
photoelectric effect, which the
wave theory cannot account for.

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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If an electron jumps one orbit closer to the
nucleus, it must emit energy equal to the
difference of the energies of the two orbits.
?
Conversely, when the electron jumps to a larger
orbit, it must absorb a quantum of light equal in
energy to the difference in orbits.

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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When a high frequency photon hits electrons,
it gets scattered. There is a decrease in
the energy of the photon. The lost energy
from the photon is transferred to the recoiling
electrons.

25
Wave behavior of the electron?
Interference pattern of a wave!

26
Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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At constant accelerating voltage. G.P. Thomson in 1928 performed
experiments with thin foil of gold in
place of nickel crystal. A diffraction
pattern is observed.
2𝑑𝑆𝑖𝑛𝜃 = 𝑛𝜆
X-rays
2𝑑𝑆𝑖𝑛𝜃 = (𝑛 + 1)𝜆
2𝑑 𝑆𝑖𝑛𝜃 − 𝑆𝑖𝑛𝜃 = 𝜆
2 × 2.15 × 𝑆𝑖𝑛62° − 𝑆𝑖𝑛31° = 𝜆
𝜆 = 0.158 nm

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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de Broglie
wavelength
The momentum of a wave
The wavelength of a particle
Matter waves
Wave-particle duality
Particle-like wave
Wave-like particle
Compton scattering
Plank’s quanta
Davisson & Germer exp.
Double slit experiment
Thomson’s gold foil exp.

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A photon
𝐸 = ℎ𝜈
2
𝐸 = mc2
Planck’s
quanta
As a particle of mass ‘m’
2
2
Electron in potential V
• For a cricket ball of mass 160 gm and
velocity of 150 km/h is λ=0.98X10-34 m.
• For an electron accelerated by applied
potential of 1 volt is λ=1.228 nm.
Mechanical waves?
Electromagnetic waves?
Pilot waves
Probability waves

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At constant accelerating voltage.
2𝑑𝑆𝑖𝑛𝜃 = (𝑛 + 1)𝜆
2𝑑 𝑆𝑖𝑛𝜃 − 𝑆𝑖𝑛𝜃 = 𝜆
2 × 2.15 × 𝑆𝑖𝑛62° − 𝑆𝑖𝑛31° = 𝜆
𝜆 = 0.158 nm
𝜆 =
.
𝑛𝑚
𝜆 = 0.155 𝑛𝑚
𝜆 =
.
𝑛𝑚
V = 60 V

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Breakthroughs
in science
Plank: Quanta
of energy (1900)
Einstein:
Photon of light
(1905)
Bohr: Atom’s
discrete states
(1913)
Compton: Particle
behavior of
radiation (1923)
De Broglie:
Dual nature
(1923)
Davisson, Germer:
Wave behavior of
electrons (1927)

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Whether electron is a particle or a wave?
Interference pattern of a wave!

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If there is no observer. If there are observers.
 Somehow, nature knows whether we have the information
of which slit the electron passed through.
 If the particle is marked in some fashion, you will not get
an interference pattern at the screen.
Does it mean that presence of
observers disturbed the outcome?

35
 Nature behaves one way when you are looking and in a completely different
way when you are not looking.
 It seems that when we observe we disturb whatever we are trying to observe.

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What does it mean to observe something?
This is a very strange result, since it seems to indicate that
the observation plays a decisive role in the event and that
reality varies, depending upon whether we observe it or not.
- Heisenberg
A lamp is required
to illuminate the
particle that enables
us to see it.
How clearly we can see depends
on the wavelength of the
illuminating light.
Smaller the wavelength, better is
the resolution.
Hence to see the electron, we
should use gamma rays
(extremely small wavelength).
The light failing on the
particle bounces off, and
reaches our eye to form
an image.
From this image we get
idea of particle’s
position.

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The Uncertainty in measurement
There is a serious problem with the gamma-ray microscope.
When we illuminate the electron with gamma rays, we are really
bombarding it with a stream of photons, which impart momentum
(Compton scattering) and disturb the position of electron.
Thus to measure the position of electron we must illuminate it; but
the moment we shine the election it is disturbed and moves away
from the position it was originally in.
Heisenberg’s conclusion:
One cannot simultaneously measure
with infinite precision, both the position
and the momentum of a particle.

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The Heisenberg Uncertainty Principle
It is impossible to simultaneously describe with absolute accuracy
the position and momentum of a particle
It is impossible to simultaneously describe with absolute accuracy
the energy of a particle and the instant of time the particle has this
energy
Δ𝐸. Δ𝑡 ≥ ℏ/2
When the Heisenberg uncertainty principle is applied to electrons it
states that we can not determine the exact position of an electron.
Instead, we could determine the probability of finding an electron at
a particular position.
Δ𝑝 . Δ𝑥 ≥ ℏ
0
Conjugate pairs
☼Heisenberg’s uncertainty principle is true for conjugate pairs only.

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Predictions or results of uncertainty principle:
1. Nonexistence of free electron in nucleus
2. Estimate of radius of Bohr’s first orbit
3. Zero point energy of simple harmonic oscillators

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1. Nonexistence of free electron in nucleus
According to theory of relativity, energy
of a particle is given by the relation
𝐸 = 𝑝 𝑐 + 𝑚 𝑐
Diameter of nucleus
~10-14 m
Maximum uncertainty in
position of electron, if it is
inside the nucleus:
xmax=10-14 m
Minimum uncertainty in
momentum of electron, if
it is inside the nucleus:
pmin= ℏ /𝑥 pmin= 1.055 × 10 𝐽𝑠/10−14 m
pmin= 1.055 × 10 𝑘𝑔 𝑚/𝑠
pmin= 1.055 × 10 𝑘𝑔 𝑚/𝑠
𝐸 = 𝑝 𝑐 + 𝑚 𝑐
𝐸 = 3.165 × 10 𝐽
m0= 9.1 × 10 𝑘𝑔
𝐸 =
3.165 × 10 𝐽
1.6 × 10 𝐶
= 19.78 𝑀𝑒𝑉
• Thus, if a free electron exists in the nucleus it
must have a minimum energy of about 20 MeV.
• The maximum K.E. of a -particle, emitted from
radioactive nuclei is of the order of 4 MeV.
Therefore electrons cannot be present within the nucleus.

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2. Estimate of radius of Bohr’s first orbit
Ze
e
x
Kinetic energy of electron
Δ𝑝 . Δ𝑥 ≥ ℏ
Δ𝑝 =
ℏ
𝑥
𝐾 =
𝑝
2𝑚 𝐾 =
(𝑝)
2𝑚
=
ℏ
2𝑚(𝑥)
Potential energy of electron 𝑉 =
(𝑍𝑒)(−𝑒)
4𝜋𝜀 𝑥
𝑉 =
(𝑍𝑒)(−𝑒)
4𝜋𝜀 𝑥
Uncertainty in total energy of electron 𝐸 =
ℏ
2𝑚(𝑥)
−
𝑍𝑒
4𝜋𝜀 𝑥
Uncertainty in total energy of
electron will be minimum if
𝜕(𝐸)
𝜕(𝑥)
= 0
−
ℏ
𝑚 𝑥
+
𝑍𝑒
4𝜋𝜀 𝑥
= 0 𝑥 =
4𝜋𝜀 ℏ
𝑚𝑍𝑒
𝑥 =
𝜀 ℎ
𝜋𝑚𝑍𝑒
Bohr’s first orbit

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Total energy of oscillator
at displacement x 𝐸 =
𝑝
2𝑚
+
𝑚𝜔 𝑥
2
xmax= 𝑎 x = 2𝑎 𝑝 =
ℏ
𝑝 =
ℏ
𝐸 = 𝐾 + 𝑉 =
𝑝
2𝑚
+
𝑚𝜔 𝑥
2
=
ℏ
8𝑚𝑎
+
𝑚𝜔 𝑎
2
Value of ‘𝑎’ at which total energy is minimum?
Is it 0?

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𝐸 =
ℏ
8𝑚𝑎
+
𝑚𝜔 𝑎
2
Let at 𝑎=‘A’ at which total energy is minimum?
𝜕𝐸
𝜕𝑎
= 0 → 𝐴 =
ℏ
2𝑚𝜔
𝐸 =
ℏ 2𝑚𝜔
8𝑚ℏ
+
𝑚𝜔
2
ℏ
2𝑚𝜔
𝐸 =
1
2
ℏ𝜔 =
1
2
ℎ𝜈

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1. Find the de Broglie wavelengths of (a) a 46-g golf ball with a velocity of 30 m/s, and
(b) an electron with a velocity of 107 m/s.
2. Calculate the de-Broglie wavelength associated with proton of kinetic energy 100
eV. (Mass of proton = 1.6725 x 10-27 kg).
3. Thermal speed of hydrogen molecule at 0° C is ~ 1.84 x 103 m/s. Find its de-Broglie
wavelength.
4. Call a photon and an electron of the same energy have the same wavelength?
5. Calculate the de-Broglie wavelength of (a) an electron accelerated by a potential
difference of 10,000 V; and (b) an electron moving with a velocity of 0.01 c, where c is
the speed of light.
6. Find the momentum of a proton whose de Broglie wavelength is 1 fm.
7. An electron has a de-Broglie wavelength of 4 pm. Find its kinetic energy and velocity.
Given h =6.63 x 10-34 Js

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8. An electron initially at rest is accelerated through a potential difference of 4900 V.
Compute its (i) momentum, (ii) the de-Broglie wavelength, and (iii) the magnitude of
wave propagation vector of the electron.
9.
10.
11.
12.
Given h =6.63 x 10-34 Js

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Matter waves
Δ𝑝 . Δ𝑥 ≥ ℏ/2
How will you define a system?
State of a system is not defined
classically but it requires a
mathematical expression named
as wave function .
We shell call this state of a system, the quantum
state. It is so formulated that all information
about the system is contained in .
Any variation in property/behaviour of the
system will be reflected in .

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While itself has no physical interpretation, its square 𝟐
at a
particular place at any time is the probability of finding the
particle there at that time.
Quantum physics
is probabilistic in
nature.
Probability cannot be negative
If 𝜓 is a complex function then 𝝍𝟐 = 𝝍∗𝝍
𝝍∗ is complex conjugate of 𝝍
╠ may be any function, real or complex, akin to the behavior
of the real physical system.
The problem of quantum mechanics is to determine for a
body when its motion is limited by the action of external forces.

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Normalization
Finite,
continuous,
Single valued
𝑑𝜓/𝑑𝑟
Finite,
continuous,
Single valued
→ 0
at r = ∞.

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► Applying certain operations/instruction on wave function information/observable
can be extracted.
► These information specific operations are called the operator for a given observable.
𝑥 ,
Eigen value equation
∗

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or
𝝏𝟐𝝍(𝒙)
𝝏𝒙𝟐
𝟐𝒎 𝑬 𝑽
ℏ𝟐
A basic physical principle that cannot be derived from anything else
It is a basic principle in itself
𝟐 𝟐
𝟐

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Postulates 1: State of a physical system: Wave function
Postulates 2: Operator corresponding to a classical observable
Postulates 3: Measurement of physical quantity: eigen value equation
Postulates 4: Probabilistic measurement: expectation value
Postulates 5: Evolution of wave function (The Schrodinger equation)
Set rules to find the solution of the problem in quantum mechanics.
𝜓∗𝑂𝜓𝑑𝑉
𝟐 𝟐
𝟐
𝑝̂𝑥 ≡ −𝑖ℏ , 𝐸 ≡ 𝑖ℏ
𝑂 𝜓 𝑟, 𝑡 = 𝑂. 𝜓(𝑟, 𝑡)

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Schrodinger equation may have a number of solutions
as the wave function. Which solution is of our use?
Superposition is the characteristic property of the waves so as of a wave function of the quantum
state of a quantum system.
A linear combination of solutions of Schrödinger’s equation for a given system is also a solution!
It is wave function, not the
probability, which add.

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Schrodinger’s cat
On opening the box 𝜓 will collapse to either 𝜓 or 𝜓 .

55
The formation of ss, pp, and
pp bonding molecular
orbitals.
linear combination of solutions of Schrödinger’s
equation for a given system is also a solution!

56
For electron A
For an electron there are two quantum states, i.e., spin-
up and spin-down states represented by ↓
and ↑
For electron B
↑
and ↓
For the combination the quantum
state will be the linear combination:
↑ ↓ ↓ ↑
for the requirement of
Pauli exclusion principle.
Entangled-state of
two quantum particles
These two electrons are related with each other via some condition,
they are entangled.
↑
and ↓

57
Quantum Entanglement
Key Characteristics:
Non-locality: Particles seem "connected"
over large distances.
Quantum Correlation: Measurement of
one particle affects the state of the other.
Quantum entanglement is the phenomenon of a
group of particles being generated, interacting, or
sharing spatial proximity in such a way that
the quantum state of each particle of the group
cannot be described independently, including when
the particles are separated by a large distance.
Spooky Science

58
Entangled State of Two Particles: Where can I find them?
• Superconductors: Cooper pair.
• Superconducting qubits: IBM's or Google's quantum
computers.
• Trapped Ions or Atoms: Using electromagnetic traps,
individual ions or atoms can be entangled by controlled
laser interactions.
• Beam Splitters: Entangled photon pairs can also be
generated using beam splitters and polarizers.

59
Aspect Entangled State Non-Entangled State
Definition
Quantum correlation
between particles.
Particles behave
independently.
Mathematical Form
Cannot be separated into
individual particle states.
Can be written as a
product of states.
Measurement Effect
Measurement of one
instantly affects the other.
Measurement of one does
not affect the other.
Entangled vs. Non-Entangled
1. Quantum Cryptography: Secure communication via Quantum Key Distribution (QKD).
2. Quantum Computing: Entangled qubits perform computations faster than classical bits.
3. Quantum Teleportation: Transfer quantum states over large distances.
4. Quantum Networking: Basis for a future quantum internet.

60
1) Free particle
Wave function No restriction = no force = no potential
or
Since k may have any value
The particle may have continuous energy.
There is no discreetness in behavior of free particle.
Unrestricted particle that may have any energy.
Unrestricted particle that may be anywhere in space.
A wave is associated with particle.
Lets introduce restriction in motion.

61
Particle’s motion is restricted
Electrons in Atoms
Quantum Dots: These are semiconductor particles that confine electrons
in three dimensions, effectively creating a "box" for the
electrons. The size of the quantum dot determines the
energy levels and the properties of the material, which can
be used in applications like lasers and displays.
Molecular Vibration: The atoms in a molecule can be thought
of as confined to specific vibrational energy levels,
leading to quantized vibrational states.
Optical Cavities: In lasers, light can be confined in a cavity,
leading to quantized modes of light. The behavior of
photons in these cavities can be analogous to
particles in a box.
2) Particle in a box
Electrons in Atoms: Electrons are often modeled as particles in a box
when considering their behavior in atoms. The potential
well created by the nucleus confines the electrons, leading
to quantized energy levels.

62
e-
𝐼 𝐼𝐼
𝜕 𝜓(𝑥)
𝜕𝑥
+
2𝑚 𝐸 − 𝑉
ℏ
𝜓(𝑥) = 0
𝜓 (𝑥) = 0
𝜓 (𝑥) ≠ 0
𝜓 (𝑥) = 0
Boundary conditions
𝜓(𝑥) 𝑑𝑥 = 1
𝜓 (𝑥) 𝑑𝑥 = 1
𝜓 (𝑥) 𝑑𝑥 + 𝜓 (𝑥) 𝑑𝑥 + 𝜓 (𝑥) 𝑑𝑥 = 1
𝜓 = 0
𝜓 = 0

63
𝜕 𝜓(𝑥)
𝜕𝑥
+
2𝑚𝐸
ℏ
𝜓(𝑥) = 0
𝑘 =
2𝑚𝐸
ℏ
𝜕 𝜓 (𝑥)
𝜕𝑥
+ 𝑘 𝜓 (𝑥) = 0
𝜓 𝑥 = 𝐴. 𝑆𝑖𝑛 𝑘𝑥 + 𝐵. 𝐶𝑜𝑠(𝑘𝑥)
𝑘 =
𝑛𝜋
𝐿
,
𝜓 (𝑥) = 𝐴𝑆𝑖𝑛
𝑛𝜋𝑥
𝐿
𝑆𝑖𝑛 𝑘𝐿 = 𝑆𝑖𝑛 𝑛𝜋
𝑛𝜋𝑥
𝐿
𝜓 = 0 = 𝐴. 𝑆𝑖𝑛 𝑘𝐿
𝜓 = 0 = 0 + 𝐵
𝐸 =
ℏ 𝑘
2𝑚
𝐸 =
ℏ 𝜋
2𝑚𝐿
𝑛
𝑛 = 0, ±1, ±2, . . .
o But if n=0, 𝑛 = ±1, ±2, . . .
𝐸 ≠ 0

64
𝑛𝜋𝑥
𝐿
𝜓 (𝑥) 𝑑𝑥 = 1
𝐴𝑆𝑖𝑛
𝑛𝜋𝑥
𝐿
𝑑𝑥 = 1
𝐴
2
1 − 𝐶𝑜𝑠
2𝑛𝜋𝑥
𝐿
𝑑𝑥 = 1
𝐴
2
𝑥 −
𝐿
2𝑛𝜋
𝑆𝑖𝑛
2𝑛𝜋𝑥
𝐿
= 1
𝐴
2
𝐿 = 1
𝐴 =
2
𝐿
𝜓 (𝑥) =
2
𝐿
𝑆𝑖𝑛
𝑛𝜋𝑥
𝐿
𝐸 =
ℏ 𝜋
2𝑚𝐿
𝑛

65
𝑛 = 1,2,3,4, . . . . .
𝜓 𝑥, 𝑡 =
2
𝐿
𝑆𝑖𝑛
𝑛𝜋𝑥
𝐿
Probabilities
Wave functions

66
Why not n = 0???
1. If a particle has zero energy, it will be at rest inside the well violating Heisenberg’s
uncertainty principle.
2. Confinement of the particle to a limited region in space requires: i.e., Δxmaximum ~ L which
leads to Δpminimum ~ ℎ/2L, causing minimum kinetic energy of the system 𝐸 =
ℏ
.
The zero-point energy reflects the necessity of a minimum motion of a
particle due to localization.
The zero-point energy occurs in all bound state potentials.
Physical consequences in microscopic world
Without zero-point motion, atoms would not be stable, for the electrons
would fall into the nuclei.
Zero-point energy prevents helium from freezing at very low temperatures.

67
3) Particle encountering a potential barrier
Semiconductors: Tunneling is essential in
tunnel diodes and transistors.
- Nuclear Fusion: Tunneling allows particles
to overcome repulsive forces in nuclear
reactions.
- Scanning Tunneling Microscopy (STM):
Utilizes tunneling to image surfaces at the
atomic level.
•Radioactive decay: Heisenberg
Uncertainty results in the probability of
particles (in nucleus) coming outside the
nucleus energy well.

68
𝑽 =
0 𝑥 ≤ 0
𝑈 0 < 𝑥 < 𝐿
0 𝑥 ≥ 𝐿
𝑥 = 0 𝐿
𝜓 𝜓 𝜓
𝑘 =
2𝑚 𝑈 − 𝐸
ℏ
𝑘 =
2𝑚𝐸
ℏ
,
𝝏𝟐𝝍(𝒙)
𝝏𝒙𝟐
+
𝟐𝒎 𝑬 − 𝑽
ℏ𝟐
𝝍(𝒙) = 𝟎
𝜕 𝜓 (𝑥)
𝜕𝑥
+
2𝑚𝐸
ℏ
𝜓 𝑥 = 0
𝜕 𝜓 (𝑥)
𝜕𝑥
+
2𝑚𝐸
ℏ
𝜓 (𝑥) = 0
𝜕 𝜓 (𝑥)
𝜕𝑥
+
2𝑚 𝐸 − 𝑈0
ℏ
𝜓 (𝑥) = 0

69
𝑘 =
2𝑚 𝑈 − 𝐸
ℏ
𝑘 =
2𝑚𝐸
ℏ
,
𝑰
𝒊𝒌𝟏𝒙 𝒊𝒌𝟏𝒙
𝑰𝑰
𝒌𝟐𝒙 𝒌𝟐𝒙
𝑰𝑰𝑰
𝒊𝒌𝟏𝒙 𝒊𝒌𝟏𝒙

70

71
𝑘2 =
1
ℏ
2𝑚 𝑈 − 𝐸
𝜓 = 𝐴𝑒 + 𝐵𝑒
Quantum mechanical
tunneling

Introduction to Quantum Mechanics (4).pdf

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